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I am trying to solve the vorticity equation, Eq. 9 in this paper:

\begin{aligned} \partial_{t} w(x, t)+u(x, t) \cdot \nabla w(x, t) &=\nu \Delta w(x, t)+f(x), & & x \in(0,1)^{2}, t \in(0, T] \\ \nabla \cdot u(x, t) &=0, & & x \in(0,1)^{2}, t \in[0, T] \\ w(x, 0) &=w_{0}(x), & & x \in(0,1)^{2} \end{aligned}

I have tried to play around with various methods, but apparently none of them is working. I think the difficulty is related to the fact that the equation contains a third order spatial derivative, and I am not sure how to proceed with this kind of problems. Does anyone know a solution to this kind of issue (or can suggest any useful reference)?

Calvin Khor
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FM87
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    what kind of solution are you looking for? I'll note that boundary conditions are missing, and the relationship $w=\text{curl}u$ is not included in Eq. 9 for some reason (and you will need the Biot-Savart law to write $u$ in terms of $w$) – Calvin Khor Feb 14 '21 at 06:56
  • Hi Calvin, I should have specified the problem better: yes, the definition of $\omega$ is included. The domain is a simple hypercube: $t, x, y \in (0,1)$. I have explicit initial conditions for $u(t=0,x,y)$, $v(t=0,x,y)$ (hence for $\omega(t=0,x,y) = \omega_0$) and I'd like to impose periodic boundary conditions in the spatial directions, namely $u(t,x=0,y) = u(t,x=1,y)$, $u(t,x,y=0) = u(t,x,y=1)$, $v(t,x=0,y) = v(t,x=1,y)$, $v(t,x,y=0) = v(t,x,y=1)$. Also, I have an explicit expression for $f(x,y)$. This is what they are using in the paper, as far as I can see. What do you think? – FM87 Feb 14 '21 at 12:20
  • OK, I don't work in numerics or machine learning, but here's what I can tell you. There exists a procedure to `construct' a solution, for any (sufficiently smooth) initial data, but maybe not for all $t\in(0,1)$. There are conditions that guarantee the solution can exist for $t\in (0,1)$. In many special cases, you can actually write down the formula for the solution $u$, but not in general. This is usually done for Navier-Stokes (in the case of the periodic box) via Galerkin approximations of $u$. I would expect you can do a similar thing directly for $\omega$. – Calvin Khor Feb 15 '21 at 02:21
  • For the Galerkin approximations, you might consider reading the book of Robinson, Rodrigo, and Sadowski "The Three-Dimensional Navier Stokes Equations", although that is in 3D, it shouldn't be very different. For some examples of explicitly solving special cases of the vorticity equation, you might want to see Bertozzi & Majda's "Vorticty and Incompressible Flow", specifically chapter 2. (this book is much more terse and requires more time to read) These assume some familliarity with pure mathematics, for the more numerics side I think Girault and Raviart's book is used, but I'm not familiar – Calvin Khor Feb 15 '21 at 02:28
  • And finally I'll explicitly note that I used "construct a solution" to mean "prove that there exists a solution", even if we know absolutely nothing for times larger than zero other than its existence (in some regularity/integrability class, e.g. twice continuously differentiable and square integrable) – Calvin Khor Feb 15 '21 at 02:29
  • Hi Calvin, thanks. This is all useful and I'll check these references. I'm not sure why this question was migrated here though, as I was asking about the possibility to solve this in Mathematica. – FM87 Feb 15 '21 at 08:20
  • I am not at all aqquainted with Mathematica.SE, and it seems reasonable to me to assume that if you ask a question in Mathematica.SE, then you are trying to do something "in Mathematica", but apparently that's not how it works? If you plan to repost there, perhaps explicitly state that you want to implement a solver in Mathematica (the current form of the question does not have this). – Calvin Khor Feb 15 '21 at 08:23
  • Also, from the paper: "The equation is solved using the stream-function formulation with a pseudospectral method. First a Poisson equation is solved in Fourier space to find the velocity field. Then the vorticity is differentiated and the non- linear term is computed is physical space after which it is dealiased. Time is advanced with a Crank–Nicolson update where the non-linear term does not enter the implicit part." So if you plan to reproduce the paper's results, this is precisely the method you should attempt. The 1st step is to use the Biot-Savart law to get $u$ (with some Fourier approx) – Calvin Khor Feb 15 '21 at 08:24
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    Hi Calvin, thanks for the advice, it was my first question so probably I did something wrong. However, the original title of my question was "Vorticity equation in Mathematica". That was changed and the question migrated. Maybe the moderator thought I had mistaken Mathematica with Mathematics :). Concerning the paper itself, we are trying to build a different solver than what they have in the paper but still we would like to use that equation as a benchmark problem, and compare with standard solvers such as Mathematica. Thanks a lot for your help anyway, you were very kind! – FM87 Feb 15 '21 at 09:19

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